Fixed point stability and decay of correlations

Vicari, Ettore and Zinn-Justin, Jean (2006) Fixed point stability and decay of correlations. New Journal of Physics, 8 . p. 321. ISSN 1367-2630

Abstract

In the framework of the renormalization-group theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective $\Phi^4$ theories, having an N-component fundamental field $\Phi_i$ and containing up to fourth-order powers of the field components. Their renormalization-group flow is usually characterized by several fixed points. We give here strong arguments in favour of the following conjecture: the stable fixed point corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent $\eta$ describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the $\epsilon$-expansion. Then, we discuss its validity beyond the $\epsilon$-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.

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